Tensor Field Networks (TFNs): 
Many methods have been proposed for using equivariant features for rotation canonicalization for 3D data, for instance, spherical CNNs [1]}, [2]}, vector neurons [3]}, or capsule networks [4]}, [5]}.
However, these and other methods [6]} are limited to 3D point cloud or voxel representations.
In this paper, we extend Tensor Field Networks [7]}, [8]} pose canonicalization directly on samples from NeRF's density field.
TFNs operate by computing Type-\(\ell \)  real-valued spherical harmonic functions that are rotation equivariant: \(D^{\ell }(R)Y^{\ell }(X) = Y^{\ell }(Rx)\) .
Where \(D^{\ell }: \text{SO}(3) \rightarrow \text{SO}(2 \ell + 1)\)  is the Wigner matrix of type \(\ell \)  (please see [7]}, [8]} for details).
For a 3D point set \(P \in \mathbb {R}^{N \times 3}\) , a rotation-equivariant TFN convolution layer (EQConv) at point \(p \in P\)  is defined as: \(\text{EQConv}^{J}(p) = Q^{(n,\ell ), J}\left(\sum _{y \in {}^{2R}\mathcal {N}} k^{n}_r(p - y) \otimes s^{\ell }[y]\right)\) .
Where \({}^{2R}\mathcal {N}\)  is a set containing neighbors of point \(p\)  at twice the resolution and \(Q^{(n,\ell ), J}(\cdot )\)  is the Clebsh Gordon decomposition to combine type-\(n\)  kernel \(k^{n}_r\)  and type-\(\ell \)  equivariant signal \(s^{\ell }\) .
The \(\text{EQConv}(\cdot )\)  layer aggregates type-\(\ell \)  feature \(s^{\ell }\)  at multiple receptive fields along with learnable weights to perform equivariant convolution.
Our method uses the EQConv layer as the fundamental building block to canonicalize continuous fields.
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